solid_state_physics/tutorial/2_04s.tex

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  30 \begin{document}
  31
  32 % header
  33 \begin{center}
  34  {\LARGE {\bf Materials Physics II}\\}
  35  \vspace{8pt}
  36  Prof. B. Stritzker\\
  37  SS 2008\\
  38  \vspace{8pt}
  39  {\Large\bf Tutorial 4 - proposed solutions}
  40 \end{center}
  41
  42 \vspace{4pt}
  43
  44 \section{Legendre transformation and Maxwell relations}
  45
  46 \begin{enumerate}
  47  \item Legendre transformation:
  48        \begin{eqnarray}
  49        dg &=& df - \sum_{i=r+1}^{n} d(u_ix_i)\nonumber\\
  50           &=& df - \sum_{i=r+1}^{n} (x_idu_i + u_idx_i)\nonumber\\
  51           &=& \sum_{i=1}^r u_idx_i - \sum_{i=r+1}^n x_idu_i\nonumber
  52        \end{eqnarray}
  53        \[
  54        \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n)
  55        \]
  56  \item Use $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
  57        $-p=\left.\frac{\partial E}{\partial V}\right|_S$.\\
  58        Start with internal energy $E=E(S,V)$:
  59        \[
  60        \Rightarrow dE=\frac{\partial E}{\partial S}dS +
  61                       \frac{\partial E}{\partial V}dV =
  62                       TdS - pdV
  63        \]
  64        Enthalpy $H=E+pV$:
  65        \[
  66        \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp
  67        \]
  68        \[
  69        \Rightarrow
  70        \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and }
  71        \left.\frac{\partial H}{\partial p}\right|_S=V
  72        \]
  73        Helmholtz free energy $F=E-TS$:
  74        \[
  75        \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT
  76        \]
  77        \[
  78        \Rightarrow
  79        \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and }
  80        \left.\frac{\partial F}{\partial T}\right|_V=-S
  81        \]
  82        Gibbs free energy $G=H-TS=E+pV-TS$:
  83        \[
  84        \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT
  85        \]
  86        \[
  87        \Rightarrow
  88        \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and }
  89        \left.\frac{\partial G}{\partial T}\right|_p=-S
  90        \]
  91  \item Maxwell relations:\\
  92        Internal energy: $dE=TdS-pdV$
  93        \[
  94        \frac{\partial}{\partial S}
  95        \left(\left.\frac{\partial E}{\partial V}\right|_S\right)_V=
  96        \frac{\partial}{\partial V}
  97        \left(\left.\frac{\partial E}{\partial S}\right|_V\right)_S
  98        \Rightarrow
  99        \left.-\frac{\partial p}{\partial S}\right|_V=
 100        \left.\frac{\partial T}{\partial V}\right|_S
 101        \]
 102        Enthalpy: $dH=TdS+Vdp$
 103        \[
 104        \frac{\partial}{\partial S}
 105        \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p=
 106        \frac{\partial}{\partial p}
 107        \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S
 108        \Rightarrow
 109        \left.\frac{\partial V}{\partial S}\right|_p=
 110        \left.\frac{\partial T}{\partial p}\right|_S
 111        \]
 112        Helmholtz free energy: $dF=-pdV-SdT$
 113        \[
 114        \frac{\partial}{\partial V}
 115        \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T=
 116        \frac{\partial}{\partial T}
 117        \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V
 118        \Rightarrow
 119        \left.-\frac{\partial S}{\partial V}\right|_T=
 120        \left.-\frac{\partial p}{\partial T}\right|_V
 121        \]
 122        Gibbs free energy: $dG=Vdp-SdT$
 123        \[
 124        \frac{\partial}{\partial p}
 125        \left(\left.\frac{\partial G}{\partial T}\right|_p\right)_T=
 126        \frac{\partial}{\partial T}
 127        \left(\left.\frac{\partial G}{\partial p}\right|_T\right)_p
 128        \Rightarrow
 129        \left.-\frac{\partial S}{\partial p}\right|_T=
 130        \left.\frac{\partial V}{\partial T}\right|_p
 131        \]
 132 \end{enumerate}
 133
 134 \section{Thermal expansion of solids}
 135
 136 \begin{enumerate}
 137  \item Coefficients of thermal expansion:\\
 138        Consider a cube with side lengthes $L_1,L_2,L_3$.
 139        Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}=
 140                             \frac{1}{L_2}\frac{\partial L_2}{\partial T}=
 141                             \frac{1}{L_3}\frac{\partial L_3}{\partial T}=
 142                             \alpha_L$.
 143        \begin{eqnarray}
 144        \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}=
 145        \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)=
 146        \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+
 147                                 L_1L_3\frac{\partial L_2}{\partial T}+
 148                                 L_1L_2\frac{\partial L_3}{\partial T}\right)
 149                                 \nonumber\\
 150        &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+
 151           \frac{1}{L_2}\frac{\partial L_2}{\partial T}+
 152           \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
 153        \end{eqnarray}
 154  \item \[
 155        dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
 156        \]
 157        \[
 158        \left.\frac{\partial E}{\partial T}\right|_V=
 159        \left.\frac{\partial E}{\partial S}\right|_V
 160        \left.\frac{\partial S}{\partial T}\right|_V=
 161        T\left.\frac{\partial S}{\partial T}\right|_V
 162        \Rightarrow
 163        \left.\frac{\partial S}{\partial T}\right|_V=
 164        \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
 165        \]
 166        \[
 167        \textrm{Using } F=E-TS \textrm{ and }
 168        TS=T\int_0^T\frac{\partial S}{\partial T'}dT'
 169        \textrm{ (Entropy density vanishes at $T=0$)}
 170        \]
 171        \[
 172        \Rightarrow
 173        p=-\frac{\partial}{\partial V}\left(
 174        E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
 175        \right)
 176        \]
 177        Harmonic approximation of the internal energy:
 178        \[
 179        E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
 180        \sum_{{\bf k}s}
 181        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
 182        \]
 183        \[
 184        \ldots
 185        \]
 186        \[
 187        x=\hbar\omega_s({\bf k})/T'
 188        \]
 189        \[
 190        \ldots
 191        \]
 192        \[
 193        \Rightarrow
 194        p=-\frac{\partial}{\partial V}\left(
 195        E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
 196        \right)+
 197        \sum_{{\bf k}s}\left(-\frac{\partial}{\partial V}\hbar\omega_s({\bf k})
 198        \right)\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
 199        \]
 200  \item The pressure depends on temperature
 201        only if the normal mode frequencies depend on the volume.
 202        However, the normal mode frequencies of a rigorously harmonic crystal
 203        are unaffected by a change in volume.\\
 204        $\Rightarrow$
 205        The pressure solely depends on the volume.\\
 206        $\Rightarrow$
 207        The pressure required to maintain a given volume
 208        does not vary with temperature.
 209        \[
 210        \left.\frac{\partial p}{\partial T}\right|_V=0
 211        \]
 212        \[
 213        \left.\frac{\partial V}{\partial T}\right|_p=
 214        -\frac{\left.\frac{\partial p}{\partial T}\right|_V}
 215              {\left.\frac{\partial p}{\partial V}\right|_T}=0
 216        \]
 217  \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
 218        and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
 219        \[
 220        C_p-C_V=\left.\frac{\partial E}{\partial T}\right|_p-
 221        \left.\frac{\partial E}{\partial T}\right|_V=
 222        \frac{\partial E}{\partial S}
 223        \left.\frac{\partial S}{\partial T}\right|_p-
 224        \frac{\partial E}{\partial S}
 225        \left.\frac{\partial S}{\partial T}\right|_V=
 226        T\left.\frac{\partial S}{\partial T}\right|_p-
 227        T\left.\frac{\partial S}{\partial T}\right|_V=
 228        T\left(
 229        \left.\frac{\partial S}{\partial T}\right|_p-
 230        \left.\frac{\partial S}{\partial T}\right|_V
 231        \right)
 232        \]
 233        Using the equality
 234        \[
 235        dS=\left.\frac{\partial S}{\partial T}\right|_p dT
 236        +\left.\frac{\partial S}{\partial p}\right|_T dp
 237        \Rightarrow
 238        \left.\frac{\partial S}{\partial T}\right|_V=
 239        \left.\frac{\partial S}{\partial T}\right|_p+
 240        \left.\frac{\partial S}{\partial p}\right|_T
 241        \left.\frac{\partial p}{\partial T}\right|_V,
 242        \]
 243        the Maxwell relation
 244        \[
 245        \left.\frac{\partial S}{\partial p}\right|_T=
 246        -\left.\frac{\partial V}{\partial T}\right|_p
 247        \]
 248        and (for a process with constant volume)
 249        \[
 250        0=dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
 251        \left.\frac{\partial V}{\partial p}\right|_T dp
 252        \Rightarrow
 253        \left.\frac{\partial p}{\partial T}\right|_V=
 254        -\frac{\left.\frac{\partial V}{\partial T}\right|_p}
 255        {\left.\frac{\partial V}{\partial p}\right|_T}
 256        \]
 257        we obtain:
 258        \[
 259        C_p-C_V=T\left(
 260        -\left.\frac{\partial S}{\partial p}\right|_T
 261        \left.\frac{\partial p}{\partial T}\right|_V
 262        \right)=T\left(
 263        \left.\frac{\partial V}{\partial T}\right|_p
 264        \left.\frac{\partial p}{\partial T}\right|_V
 265        \right)=T\left(
 266        \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
 267        {-\left.\frac{\partial V}{\partial p}\right|_T}
 268        \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
 269        TVB\alpha_V^2
 270        \]
 271        For a rigorously harmonic potential $C_p=C_V$.
 272 \end{enumerate}
 273
 274 \end{document}